Is there any thing specific about the way ASIC miners handle work that is tightly tied to the Bitcoin protocol, as opposed to some other protocol implantation that uses SHA-256 and nonce based POW? Aka, could some vastly different protocol with entirely different source code from Bitcoin still benefit from the established hashing power worldwide.
There is nothing specific about ASIC hardware that ties it to the protocol. However there is something specific that ties it to only be able to perform the SHA256d (SHA256 double) hash. The hardware is literally hard wired to only be able to perform a SHA256d hash function on whatever data it is fed.
Does the Bitcoin protocol REALLY lack that much at a fundamental level, or could it be added to and forked to retain supremacy? Is there room for another protocol to do things differently and end up with a more adopted/useful blockchain?
First of all, the protocol is different from the consensus rules. The protocol is simply just the formatting of data and how it is transmitted. What people actually care about are the consensus rules, and those are what define what is valid and not. The consensus rules can be changed, protocol rules not really. Changing the consensus rules will typically involve a fork.
Do you think this is a likely scenario? And if so, do you think the new protocol would be intentionally designed to be backwards compatible with existing mining gear, or would it be intentionally designed such that it required new mining hardware.
There are guaranteed to be changes to the consensus rule. Most of the BIPs involve some sort of change to those rules. The only thing that would require a change of mining gear is to change the algorithm used for mining. This would only happen if SHA256d was found to be unsafe and broken.